Begell House Inc.
International Journal for Multiscale Computational Engineering
JMC
1543-1649
4
1
2006
Preface: To the Memory of Professor Yuanxian Gu (1954 - 2005)
1-2
Variational Principle and Mechanical Computation for Energy Bands of Periodic Materials
3-18
Hongwu
Zhang
Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, P. R. China
Q.
Gao
Department of Engineering Mechanics, State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116024, P. R. China
Wanxie
Zhong
Dalian University of Technology
Based on the Bloch theorem and tight-binding theory, a variational principle is applied to analyze the energy bands of crystals. The stiffness matrix used in the finite element method (FEM) is introduced for the expression of the energy of the unit cell of the crystal, and thus the coordinate transformation technique in FEM is applied in the assembly of the total energy and the stiffness matrix of the crystal. The periodical boundary conditions are given, and the energy bands of the three-dimensional crystal are computed. Using the dynamic substructure model and introducing the dual variables, the energy band analysis of the free vibration of the atomic chain is transformed into symplectic eigenvalue problems. The potential energy and mixed energy are computed by combining segments recursively until the shortest periodical length of the chain is assembled. Finally, the pass-band eigenvalues of the energy bands are calculated using the Wittrick-Williams algorithm. The numerical results are given to illustrate the potential of the theory and algorithm developed.
Elastic Softening and Stiffening of Metals Surfaces
19-28
L. G.
Zhou
Department of Mechanical, Aerospace & Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA
Hanchen
Huang
Rensselaer Polytechnic Institute
Materials surfaces, which dominate nanostructures such as nanoplates, are elastically different from their bulk counterparts. This paper reports elastic softening and stiffening of metals surfaces based on a combination of analytical formulations, molecular statics calculations, and ab initio calculations. The metals in this study include face-centered-cubic Cu, body-centered-cubic W, and hexagonal-close-packed Ti. Two mechanisms dictate the surface elasticity. The bond loss of surface atoms leads to surface softening, and the accompanying bond saturation leads to surface stiffening. For close-packed Cu and Ti the percentage of missing bonds is low, and the bond loss can be compensated for by the bond saturation, leading to possible surface stiffening. On the other hand, open-structure W is unlikely to have stiffer surfaces because each missing bond constitutes a large percentage of the total coordination. In general, a metal surface can be elastically softer or stiffer than the bulk, depending on the competition of bond loss and bond saturation at the surface.
Multiscale Total Lagrangian Formulation for Modeling Dislocation-Induced Plastic Deformation in Polycrystalline Materials
29-46
Xinwei
Zhang
Civil & Environmental Engineering Department, University of California, Los Angeles (UCLA), 5731G Boelter Hall, Los Angeles, CA 90095, USA
Shafigh
Mehraeen
Civil & Environmental Engineering Department, University of California, Los Angeles (UCLA), 5731G Boelter Hall, Los Angeles, CA 90095, USA
Jiun-Shyan
Chen
Civil & Environmental Engineering Department, University of California, Los Angeles (UCLA), 5731G Boelter Hall, Los Angeles, CA 90095, USA
Nasr M.
Ghoniem
Mechanical and Aerospace Engineering, University of California, Los Angeles, Los Angeles, CA 90095, USA
Multiscale mathematical and computational formulation for coupling mesoscale dislocation mechanics and macroscale continuum mechanics for prediction of plastic deformation in polycrystalline materials is presented. In this development a total Lagrangian multiscale variational formulation for materials subjected to geometric and material nonlinearities is first introduced. By performing scale decomposition of kinematic variables and the corresponding dislocation kinematic variables, several leading-order equations, including a scale-coupling equation, a mesoscale dislocation evolution equation, and a homogenized macroscale equilibrium equation, are obtained. By further employing the Orowan relation, a mesoscopic plastic strain is obtained from dislocation velocity and its distribution, and a homogenized elastoplastic stress-strain relation for macroscale is constructed. The macroscale, mesoscale, and scale-coupling equations are solved interactively at each macroscopic load increment, and information on the two scales is passed through the macroscale integration points. In this multiscale approach the phenomenological hardening rule and flow rule in the classical plasticity theory are avoided, and they are replaced by a homogenized mesoscale material response characterized by dislocation evolution and their interactions.
Multiphysical Modeling and a Transscale Computation of Composite Materials and Their Interfaces
47-70
Li-Qun
Cao
State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Science-Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O.Box 2719, Beijing, 100080, China
Chong-Yu
Wang
Department of Physics, Tsinghua University, Beijing, 100084, China
Jun-Zhi
Cui
State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Science-Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O.Box 2719, Beijing, 100080, China
In this paper, we shall discuss kinetic equations of composite materials and their interfaces, which is a typical multiphysical problem, due to the existence of very thin interfaces. A multiphysical model and a transscale computational method will be presented. Our method is composed of four parts: the first one is the first-principle calculation (e.g., TB-DFT method) for calculating the total energy and the elastic constants of interfaces ; the second one is the homogenization method for evaluating the effective elastic constants of macroscopic structures of composite materials; the third one is the multiscale method for computing the stress field of macroscopic structures; and the final one is the quantum molecular dynamical method (e.g.,TB-MD) for calculating the local stress field of the interfaces. Some numerical results are reported. The rigorous proofs of some main theoretical results will be given in the Appendixes.
A Constitutive Model for Nanomaterials Based on Spatial Secant
71-94
Dong
Qian
Department of Mechanical, Industrial and Nuclear Engineering University of Cincinnati, Cincinnati, OH 45221-0072
Rohit H.
Gondhalekar
Department of Mechanical, Industrial and Nuclear Engineering University of Cincinnati, Cincinnati, OH 45221-0072
In this paper a finite deformation hierarchical model based on the concept of spatial secant is presented. Motivated by the fundamental differences between the definitions of continuum in the classical sense and discrete atomic structure studied in this paper, this model proposes a systematic description for the mechanics of nanomaterials with regular lattice structures. Although the proposed model is similar in its form to the crystal elasticity model, it distinguishes itself from the classical continuum model in that it directly considers the finite size effect of the atomic bond by introducing the concept of spatial secant. As a result, the proposed model is consistently linked to the mechanics of the underlying atomic structure by deriving directly from the interatomic potential with the proposed deformation measure. In contrast, it is shown that the crystal elasticity model based on the deformation gradient (or equivalently, the spatial secant) is not suitable for describing the mechanics of nanostructures due to its inability to account for the finite size effect of the interatomic bond. Following the presentation of the model, the numerical procedure to solve the constitutive model based on spatial secant is described. The implementation of the model in a typical mesh-free Galerkin formulation is illustrated in a set of benchmark problems involving two-dimensional nanostructures. The robustness and accuracy of the proposed model are shown by directly comparing the results to those obtained from full-scale atomistic simulations for the same benchmark problems. The erroneous results obtained from the spatial tangent-based crystal elasticity model highlight the importance of the proposed model for resolving the mechanics of nanostructures.
Multiscale Analysis of Adiabatic Shear Bands in Tungsten Heavy Alloy Particulate Composites
95-114
Romesh
Batra
Virginia Polytechnic Institute
B. M.
Love
Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061
We use a multiscale approach to analyze adiabatic shear bands in a tungsten heavy alloy particulate composite deformed in plane strain tension at a nominal strain rate of 5000/s. Fifty μ;m diameter circular tungsten particulates are assumed to be randomly distributed and perfectly bonded to the nickel-iron matrix. The volume fraction of particulates equals 50%. We first analyze transient coupled thermo-mechanical deformations of a homogenized body with values of thermophysical material parameters equivalent to those of the particulate composite. Time histories of deformation variables on the bounding surfaces of the centrally located 2 mm × 2 mm subregion of the 10 mm × 10 mm region are recorded. Boundary conditions of surface tractions and temperature rather than of velocities and temperature are then used to analyze plane strain coupled thermomechanical deformations of the 2 mm × 2 mm composite in which tungsten particulates are randomly distributed in the central 1 mm × 1 mm subregion of the 2 mm × 2 mm region with the remaining part comprised of the equivalent homogeneous material of the 10 mm × 10 mm body. It is found that the multiscale analysis of the problem gives an adiabatic shear band initiation time of ∼ 22 μ;s as compared to ∼ 58 μ;s in the equivalent homogenized body and ∼ 50 μ;s in the macroanalysis of deformations of the 1 mm × 1 mm region containing a randomly distributed 50% volume fraction of 50 μ;m diameter tungsten particulates.
Molecular Physical Mechanics and Multiphysics-Scale Studies
115-126
Chun
Tang
Institute of Nano Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China
Wanlin
Guo
Institute of Nano Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China
When size goes down to nanoscale or molecular scale, besides the widely known spatial and temporal multiscale problems, new physical properties and coupling effects between mechanical motion and physical, biochemical properties may become dominant, and multiphysics-scale problems arise. Under the interdisciplinary framework that combines continuum mechanics and quantum mechanical theory, here we present several selected problems, including nanointelligent behaviors, the energy dissipation mechanism, the temperature treatment strategy, bond switching, and ion channels, among others, to demonstrate some of the most common concerns in molecular physical mechanics. The examples mainly cover the range of multidisciplines from classical mechanics to quantum mechanics, etc., although for more complex phenomena such as in life science the multiphysics scale may not stop at quantum mechanics. The results show that except for the size scale and the time scale, the physical scale is also spanned in these multiscale investigations. The molecular physical mechanics needs to extend to a larger range of multiphysical scale when research goes deeper into the nature of molecular systems, certainly with greater challenges, but more arresting findings.
From Immersed Boundary Method to Immersed Continuum Methods
127-146
X. Sheldon
Wang
Midwestern State University
The objective of this paper is to present an overview of the newly proposed immersed continuum method in conjunction with the traditional treatment of fluid-structure interaction problems, the immersed boundary method, the extended immersed boundary method, the immersed finite element method, and the fictitious domain method. In particular, the key aspects of the immersed continuum method in comparison with the immersed boundary method are discussed. The immersed continuum method retains the same strategies employed in the extended immersed boundary method and the immersed finite element method, namely, the independent solid mesh moves on top of a fixed or prescribed background fluid mesh, and employs fully implicit time integration with a matrix-free combination of Newton-Raphson and GMRES iterative solution procedures. Therefore, the immersed continuum method is capable of handling compressible fluid interacting with a compressible solid. Several numerical examples are also presented to demonstrate that the proposed immersed continuum method is a good candidate for multiscale and multiphysics modeling platforms.
Effect of Nonlinear Interface Debonding on the Constitutive Model of Composite Materials
147-168
H.
Tan
Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
C.
Liu
MST-8, Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Y.
Huang
Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Philippe H.
Geubelle
Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Interface debonding, which plays an important role in the deformation and fracture of composite materials, can be characterized by a cohesive law. We use a nonlinear cohesive law for the particle/matrix interface obtained from experiments to study the effect of interface debonding on the macroscopic behavior of composite materials. The dilute solution is obtained for a composite with spherical particles subject to interface debonding and remote hydrostatic tension. For a composite with a fixed particle volume fraction, particle and matrix properties, and interface cohesive law, different particle sizes may lead to very different macroscopic behaviors, such as hardening of the composite for small particles, softening for medium particles, and unloading for large particles. Two critical particle sizes separating these three scenarios are identified. The composite with particles of the same size as well as the bimodal distribution of particle size is studied, with a focus on the effects of particle size and cohesive energy of the particle/matrix interface. For medium or large particles, the particle/matrix interface may undergo catastrophic debonding, i.e., sudden, dynamic debonding, even under static load.
Trans-scale Coupling in Multiscale Simulations
169-182
Feng
Rong
LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, P. R. China
Haiying
Wang
LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, P. R. China
Mengfen
Xia
LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, P. R. China
Fujiu
Ke
LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, P. R. China
Yilong
Bai
LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, P. R. China
Trans-scale coupling plays a significant role in multiscale problems. Since the mechanisms governing the trans-scale coupling vary from case to case, to identify and characterize the governing mechanisms of trans-scale coupling are the most crucial points in multiscale simulations. The failure of solid media is a typical multiscale process. This paper chooses two model problems, i.e., damage localization in spallation of an Al alloy and the catastrophe transition in a rock under quasi-static loading, to illustrate the trans-scale coupling in different phases of material failure. In the spallation process the governing mechanism of trans-scale effects is the coupling and competition between dynamics at different levels, which can be effectively characterized by two imposed Deborah numbers. In the catastrophe failure of heterogeneous media the governing mechanism of trans-scale coupling is the strong and sensitive coupling between the nonlinear dynamics and the disordered heterogeneity. In addition, the inverse cascade of damage evolution magnifies the effects of microstructures on failure and induces trans-scale sensitivity. Although the concept of critical sensitivity seems to be promising in catastrophe prediction, novel concepts and numerical schemes are still badly needed.
Nonuniformity Effect of Surface-Nanocrystalline Materials in Nanoindentation Test
183-196
Yueguang
Wei
LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China
Xiaoliang
Chen
LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China
Siqi
Shu
LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China
Chen
Zhu
LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China
In the present research, microstructures of the surface-nanocrystalline Al alloy material are observed and measured based on the transmission electron microscopy (TEM) technique, and the corresponding mechanical behaviors are investigated experimentally and theoretically. In the experimental research, the nanoindentation test method is used, and the load and microhardness curves are measured, which strongly depend on the grain size and grain size nonuniformity. Two kinds of the nanoindentation test methods are adopted: the randomly selected loading point method and the continuous stiffness method. In the theoretical modeling, based on the microstructure characteristics of the surface-nanocrystalline Al alloy material, a dislocation pile-up model considering the grain size effect and based on the Mott theory is presented and used. The hardness-indent depth curves are predicted and modeled.
Homogenization Method Based on Eigenvector Expansions
197-206
Jinmei
Tian
The Solid Mechanics Research Center, Beijing University of Aeronautics & Astronautics, Beijing 100083, China
Dechao
Zhu
The Solid Mechanics Research Center, Beijing University of Aeronautics & Astronautics, Beijing 100083, China
Wenjian
Xie
The Solid Mechanics Research Center, Beijing University of Aeronautics & Astronautics, Beijing 100083, China
On the basis of the eigenvector expansions, in the present paper a homogenization method is presented to evaluate the macromechanical properties of any kind of woven fabric composites. In this homogenization method, there are two kinds of finite elements with different scales. Different from the conventional homogenization method, which evaluates the homogenized elastic moduli for a heterogeneous unit cell, the present homogenization method evaluates the homogenized stiffness matrix of the heterogeneous unit cell of composite materials directly based on the eigenvector expansions, and in the homogenized stiffness matrix the diagonal elements are different. The advantage of doing it in this manner is that the homogenized stiffness matrix can depict the local geometry and material architecture within the unit cell in much more detail than the overall homogeneous elastic moduli. Two numerical examples of three-dimensional orthogonal woven fabric composites are given to illustrate the effectiveness of the method and to compare the results obtained by both methods. The first example is about the comparisons between the stiffness matrix obtained by the present homogenization method and that by the conventional homogenization method. The second example is about the comparisons among the frequencies by three different methods. Since the finite element method is adopted during numerical analysis, it is easy to extend the applications of this method to any kind of composite materials with more complicated geometry and material architecture.